Explaining the Determinant

Date: 11/16/97 at 21:52:11
From: Jeremy Carlino
Subject: Explaining the determinant
I am trying to understand what the determinant of a matrix actually
is. I have a degree in mathematics and am currently teaching Algebra
II to gifted and talented students. I know how to find the determinant
and how to teach the process of finding the determinant, but I haven't
been able to explain what it is. Please help.
Interestingly, I found in a History of Math text, <i>From Five Fingers
to Infinity,</i> that before Auther Cayley "created" matrix theory,
Liebniez and a Chinese or Japanese mathematician simultaneously and
independently discovered the determinant. How could the determinant
have been discovered before the matrix? I guess I have two questions.
Thanks in advance for any help.
Jeremy C.

Date: 11/17/97 at 19:57:10
From: Doctor Tom
Subject: Re: Explaining the determinant
Hello Jeremy,
I always think of it geometrically.
Let's look in two dimensions, at the determinant of the following:
| x0 y0 | = x0*y1 - x1*y0
| x1 y1 |
Now imagine the two vectors (x0, y0) and (x1, y1) drawn in the x-y
plane from the origin. If you consider them to be two sides of a
parallelogram, then the determinant is the area of the parallelogram.
Well, not exactly the area, the "signed" area, in the sense that if
you sweep the area clockwise, you get one sign, and the opposite sign
if you sweep it in the other direction. It's just as useful a concept
as considering area below the x-axis as negative in your calculus
course.
Swapping the vectors swaps the sign, in the same way that swapping the
rows of the determinant swaps the sign.
In one dimension, the determinant is just the number, but if you
"plot" that number on a number line, it's the (signed) length of the
line. If it goes in the positive direction from the origin, it's
positive, and negative otherwise.
In three dimensions, consider three vectors (x0,y0,z0), (x1,y1,z1),
and (x2,y2,z2). If you draw them from the origin, they form the
principle edges of a parallelepiped, and the determinant of:
| x0 y0 z0 |
| x1 y1 z1 |
| x2 y2 z2 |
is the volume of that parallelepiped.
In higher dimensions, its just the 4D (or 5D, or 6D ...) signed
"hypervolumes" of the hyper-parallelepipeds.
With this view, it's easy to see why the determinant's properties make
sense. Swapping two rows changes the order of sweeping out the volume,
and will hence turn a positive volume to negative or vice-versa.
Multiplying all the elements of a row by a constant (say 2) stretches
the parallelepiped by a factor of 2 in one direction, and hence
doubles the volume.
Adding a row to another just skews the parallelepiped parallel to one
of its faces, and hence (Cavallari's principle) leaves the volume
unchanged. (If you can't see this, plot it in two dimensions for a
couple of examples.)
Check the other allowed determinant manipulations to see how they
relate to the geometry.
Because a determinant is a fundamental geometric property of a
collection of N N-dimensional vectors, it's not too surprising that
different folks would stumble across it, even without knowing what a
matrix is.
I hope this helps.
-Doctor Tom, The Math Forum
Check out our web site! http://mathforum.org/dr.math/